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t, s
Fig. 6.25 Unit-.step response (Iateral-directional) of the general aviation airplane:
roll angle.
where
Art, -. (mi - bi CyB)(lxi Cn8" + Ixzl Cl8.) + C).8a (Ixzl Ci/)bi + Ixi CnBbi)
(6.249)
B?/, = -Cyp (Ixzl Cn8o + Ixzl q8.) + (ttll - b] Cyh)(Cl8o Cnpbi - Cn8" C,pbi)
+ Cypbi ( - Cn8,CtBbl + Cn:a CnBbi) + Lzi c. ip + qp Cnpb~
- CnBCtpb~ + Ixl Cnp (6.250)
Cv, = -Cypbi (CI&,Cnp - Cn&; Clp) + bi Cyp (Cnp Cl8" - Cip Cn8,,)
+ C).* (-C"8" + CiBbi + Ct8" Cnpbi) + bi (Ctp Cnp - Ctp Cnp)
Dp = Cv4(Cnp Ciaso - Cip Cnt,)
(6.251)
(6.252)
AIRPLANE RESPONSE AND CLOSED-LOOP CONTROL 585
The transfer function for the bank angle is given by
A4(s)
A8a~s~ =
(mis - bi C),Bs - C),p) Cy8tr mis - bi CyrS
-(Cip + bi CWs) Cl8a - (qrbis + IxzIS2)
-(Cnp + bi C,,ps) Cn8a - (CnrbiS - IzIS2)
Alar
(6.253)
Let N4,& denote the determinantin the nurnerator. Expanding, we get
6.8" = A6S2 + B6s + C4 (6.254)
where
A4 = (mi - bi Cyp) (I.zi Cn8 + Iz I CL8,) + Cy8"(Lzl C"Bbi + Iz] CLBbi)
(6.255)
Bi = -C).p (Ixzi Cn8a + Izl Clb,) + (lTII ~ bi CyB)(Cn8,,Clr - CLai Cnr)bi
- Cy8,, (CiBCnrb~- - Cip/zl - Cnp Ixzl - CnBCtrb~) .
+ (mi - bi C},r)(q8i Cnpbl - Cna Ctpbi) (6.256)
C6 = (mi - bi C).r)(Cia:,Cnp - Cn&Ctp) (6.257)
Substituting the mass and aerodynamic characteristics of the general aviation air-
plane, we get the transfer functions of the general aviation airplane as follows:
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PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL3(128)
